MSc_Eng_-Steady Three-dimensional Shock Wave Reflection Transition Phenomena

7/30/2019 MSc_Eng_-Steady Three-dimensional Shock Wave Reflection Transition Phenomena

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STEADY THREE-DIMENSIONAL SHOCK WAVE

REFLECTION TRANSITION PHENOMENA

Jeffrey Baloyi

A research report submitted to the Faculty of Engineering and the Built Environment,

of the University of the Witwatersrand, in partial fulfilment of the requirements for

the degree of Master of Science in Engineering.

Johannesburg 2008


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Formal declaration

I declare that this research report is my own unaided work. It is being

submitted for the Degree of Master of Science in Engineering to the Universityof the Witwatersrand. I also declare that all references or sources cited in this

research report were properly acknowledged. I also declare that the work

undertaken for this research report was done while I was under the employ of

the Council for Scientific and Industrial Research (CSIR).

..

Jeffrey Baloyi

.. day of .. year .

Day month year


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Abstract

Shockwave reflection has in recent times been investigated as a three-

dimensional phenomenon where geometrical effects on the reflection patternshave been given more attention than previously. A typical example is that of a

supersonic body flying over a ground plane, in which the bow wave reflects off

the ground surface. Depending on the Mach number, the reflection can be

regular below the body, but will then make a transition to the three-shock

Mach reflection pattern at some lateral position. In this report symmetrically

arranged wedges with a finite span (i.e. one above the other) were modeled

and visualised in CFD in order to investigate the three-dimensional steady

state transition from regular reflection to Mach reflection. This follows on the

work done by Skews (2000) where it was observed from shadowgraph

pictures that there seems to be a sudden jump at the transition point in the

growth of the Mach stem.

Contrary to what was observed by Skews (2000), the transition was found to

be gradual and smooth in the current CFD simulations. High visual clarity from

the CFD results could not be achieved, even after successive grid refinements

were performed on and around the shockwaves, because of the averaging

technique of fluid property values in cells performed by CFD codes. The flows

in the vicinity of the transition are examined, with particular attention to the

shear layers that are generated from the triple point lines. Because of the

inclination of the Mach stem surface to the oncoming flow the Mach number

behind this surface can be supersonic, in contrast to the two-dimensionalcase.

The steady state reflection phenomenon where there is transition from Mach

reflection to regular and then back to Mach reflection when moving laterally

outward from the vertical symmetry plane was also investigated using the

same CFD setup, but with a much wider wedge span. This particularly

interesting situation suggests the existence of complex transition criteria. The


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aim was to reproduce numerically this phenomenon observed experimentally

by Ivanov et al. (1999), and to see if these results can be replicated for a

lower Mach number attainable using a local wind tunnel. Both aims were

achieved, but with the same limitation mentioned above of the averaging

technique of fluid property values by CFD codes. There are currently no

analytical criteria for the prediction of shock wave reflection transition in the

three-dimensional case, nor for the possible existence of a dual solution

domain, as exist for two-dimensional flows. Parametric studies of the type

discussed in this report should lead to a fuller understanding of the flow

conditions of importance.


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Acknowledgements

I firstly would like to thank my supervisor, Prof. B W Skews, for his guidance

and support throughout the time I had been registered as a student. Secondly

I would also like to thank the National Research Foundation (NRF) for the

funding granted to me through my supervisor. Last and not least I would also

like to thank the Council for Scientific and Industrial Research (CSIR) for

making time available for me to be able to do my research work while being

employed by them, and also making their computer resources available for

use for my research.


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Table of contents

Formal declaration ............................................................................................ i

Abstract............................................................................................................ ii

Acknowledgements......................................................................................... iv

Table of contents ............................................................................................. v

List of figures................................................................................................... vi

Nomenclature................................................................................................... x

Introduction ......................................................................................................1

Background and literature review.....................................................................8

Objectives......................................................................................................13

Method and procedure...................................................................................14

Case descriptions.......................................................................................14

Single transition Case.............................................................................15

Double transition case ............................................................................16

Discussion of results ......................................................................................19

Double transition Case...............................................................................19

Single transition case .................................................................................32

Conclusions ...................................................................................................48References.....................................................................................................49

Appendix A: Additional pictures for the double transition case.......................51

Appendix B: Additional pictures for the single transition case........................53

Appendix C: Detailed dimensions of the wedge setup for each test case......55


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List of figures

Figure 1: Regular reflection in steady state flow. Physical (left) and shock

polar (right). Adapted from Hornung (1986)..............................................2

Figure 2: Mach reflection in steady state flow. Physical (left) and shock polar

(right). Adapted from Hornung (1986).......................................................2

Figure 3: A shock polar diagram showing the condition at which the shock

wave angle is the von Neumann angle. ....................................................4

Figure 4: A shock polar diagram showing the condition at which the shock

wave angle is the detachment angle (solid line) or the sonic angle

(dashed line).............................................................................................5

Figure 5: The von Neumann and detachment criteria as functions of Mach

number for a specific-heat ratio of 1.4, where d is the detachment

criterion angle, N is the von Neumann criterion angle and is the

incident shock wave angle. .......................................................................7

Figure 6: An oblique shadowgraph picture showing transition from regular

reflection to Mach reflection, taken by Skews (2000)................................8

Figure 7: Conventional shadowgraph showing regular reflection. Adapted from

Brown and Skews (2004)..........................................................................9Figure 8: Orthogonal shadowgraph corresponding to figure 7. Adapted from

Brown and Skews (2004)..........................................................................9

Figure 9: Laser sheet images at different span-wise positions ( z) taken by

Ivanov et al. (1999). ................................................................................10

Figure 11: shadowgraph pictures showing the absence of influence on the

Mach stem height by downstream flow conditions. Adapted from Chpoun

and Leclerc (1999)..................................................................................12

Figure 12: Pictures showing arrangements of the two wedges used in the

Single transition case..............................................................................16

Figure 13: Pictures showing arrangements of the two wedges used in the

Double transition case. ...........................................................................17

Figure 14: CFD pictures at span-wise positions (z) of, from left to right 0 mm,

158 mm and 198 mm, obtained using Fluent for M = 4.0......................21

Figure 14: Laser sheet Images at different span-wise positions ( z) taken by

Ivanov et al. (1999) for M = 4.0. ............................................................21


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Figure 16: CFD pictures at span-wise positions (z) of, from top to bottom 0

mm, 158 mm and 198 mm, obtained using Fluent for M = 3.1..............22

Figure 17: CFD pictures at span-wise positions (z) of, from top to bottom 0

mm, 158 mm and 198 mm, obtained using Fluent for M = 2.9..............23

Figure 18: A flooded plan view of Mach number plotted on the horizontal

symmetry plane. This is for M = 3.1. .....................................................24

Figure 19: A flooded plan view of pressure plotted on the horizontal symmetry

plane. This is for M = 3.1.......................................................................25

Figure 20: A plan view plot of a slip surface (shear layer) with streamlines

moving from right to left plotted on the horizontal symmetry plane, with

the wedge in the background. This is for M = 3.1..................................26

Figure 21: A plan view plot of entropy on the horizontal symmetry plane. This

is for M = 3.1. ........................................................................................27

Figure 22: A front view of Mach=2.8 isosurface showing the double transition

phenomenon. This is for M = 2.9...........................................................28

Figure 23: A side view of Mach number contours on the vertical symmetry

plane for M = 3.1...................................................................................29

Figure 24: Starccm+ images, for the halves of the two wedges, of an

isosurface of Mach 2.9, for M = 3.1.......................................................30

Figure 25: Starccm+ images, for half a wedge, of an isosurface of Mach 1,

showing the shape of the slip surface for M = 3.1. ................................31

Figure 24 shows the shape of the double transition phenomenon, but in this

instance for M = 3.1. The associated slip stream surface is shown in

figure 25..................................................................................................31

Figure 26: An oblique shadowgraph picture showing transition from regular

reflection to Mach reflection, taken by Skews (2000)..............................33

Figure 27: A 3-D Starccm+ projected view of a Mach number isosurface at

Mach = 2.9, showing transition from regular reflection to Mach reflection.

................................................................................................................33

Figure 28: A 3-D Starccm+ projected view of a Mach number isosurface at

Mach = 2.9, showing transition from regular reflection to Mach reflection.

Simulation ran with a laminar flow. .........................................................34

Figure 29: A close up of the transition from regular reflection to Mach

reflection for figure 26. ............................................................................34


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Figure 30: A close up of the transition from regular reflection to Mach

reflection for figure 27. ............................................................................35



As mentioned earlier figure 31 is a flooded plot of Mach number plotted on the

vertical symmetry plane, whereby Starccm+ is used as the CFD solver.36

Figure 32: A flooded plan view of Mach number plotted on the vertical


Figure 33: A flooded plan view of density plotted on the horizontal symmetry


Figure 34: Orthogonal shadowgraph showing the single transition

phenomenon. This picture is the same figure 8, but rotated by 180for

easy comparison with figure 29. Adapted from Brown and Skews (2004).

................................................................................................................38

Figure 35: A flooded plan view of density plotted on the vertical symmetry


Figure 36: A flooded plan view of pressure plotted on the vertical symmetry

plane. For M = 3.1.................................................................................40

Figure 37: A front view of Mach=2.9 isosurface showing the right half of the

single transition phenomenon with the vertical plane of symmetry to the

left. This is for M = 3.1...........................................................................40

Figure 38: A close-up of Figure 37.................................................................41

Figure 39: A front view of Mach=2.9 isosurface showing the lower half of the

single transition phenomenon below the horizontal plane of symmetry.

This is for M = 3.1. ................................................................................42

Figure 40: A plan view showing streamlines paths being affected by going

through shock waves. For M = 3.1. .......................................................43

Figure 41: A plan view showing streamlines paths being affected by going

through shock waves, for both sides of the wedge. For M = 3.1. ..........44

Figure 42: A side view showing streamlines paths being affected by going

through shock waves. For M = 3.1. streamlines released just below the

horizontal plane of symmetry..................................................................45

Figure 43: A front view showing streamlines paths being affected by going

through shock waves. For M = 3.1. .......................................................46


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Figure 44: An isometric view showing streamlines paths being affected by

going through shock waves. For M = 3.1. .............................................47


symmetry plane. This is for M = 2.9 ......................................................51

Figure 45: A flooded plan view of Pressure plotted on the horizontal symmetry


Figure 46: A contoured side view of Pressure plotted on the vertical symmetry

plane superimposed with the slip surface, showing the shape of the slip

surface viewed from the side. This is for M = 3.1 ..................................52

Figure 47: A flooded plan view of Pressure plotted on the horizontal symmetry

plane.......................................................................................................53

Figure 48: A side view showing streamlines (in colour) paths being affected by

going through shock waves. For M = 3.1. Streamlines released just

below the horizontal plane of symmetry..................................................54

Figure 49: A front view showing streamlines paths (in colour) being affected

by going through shock waves. For M = 3.1..........................................54


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Nomenclature

2-D Two-dimensional

3-D Three-dimensional

d detachment criterion angle

N von Neumann criterion angle

M Free stream Mach number

CFD Computational fluid dynamics

GB GigaByte

RAM Random Access Memory

Fluent Commercial CFD software

Starccm+ Commercial CFD software


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Introduction

When an object moves through a gas (for instance air) regardless of whether

the gas is stationary or has a velocity of its own, there are pressure pulses

emanating from the gas particles immediately on the surface of the object in

motion. These pressure pulses travel from the surface of the object at the

speed of sound in all directions. This is a mechanism by which a gas

continuum is aware of the presence of a solid object. Thus if one were to

visualise the streamlines of the gas continuum using smoke for instance, one

would see streamlines bent close to the object and further away from the

object.The above scenario is true for a gas if its speed relative to the object is below

the speed of sound, which is the speed at which the pressure pulses travel.

For a scenario where the speed of the gas relative to the object is faster than

the speed of sound, the gas continuum is no longer getting information from

the pressure pulses about the presence of the object, hence cannot get out of

the way of the solid object. Given that gas particles cannot simply permeate

through the solid surface, nature deals with this conundrum by creating shock

waves which bend the gas continuum around the object. Shock waves are

discontinuities in the gas continuum, because their thicknesses are in the

micrometer region and gas properties are discontinuous from one side of the

shock wave to the other side as the gas goes through the shock wave.

Now these shock waves that get generated off the surfaces of objects can be

understood as being infinitely thin surfaces that can be plane or curved.

These shock waves expand outwardly in their breadth and length; hence they

can encounter other objects in the gas continuum. When shock waves

encounter surfaces they reflect off the surface, and the pattern of their

reflection is dependent on their incoming angle, the flow speed, the boundary

layer parameters and the orientation of the reflecting surface.

There are generally two types of reflection patterns, namely: Regular

reflection where the shock wave reflection can be likened to that of a light


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beam bouncing off a flat surface, and Mach reflection where the incident or

incoming shock wave does not come into contact with the reflecting surface,

but the incident shock wave seems to bounce off just above the surface with a

Mach stem (surface in 3D) developing between the triple point (line in 3D) and

the reflecting surface. The triple point is where the incident, reflected, and

Mach stem shocks meet. The regular reflection pattern is represented in figure

1, whereas the Mach reflection pattern is represented in figure 2.

Figure 1: Regular reflection in steady state flow. Physical (left) and shock polar (right). Adapted

from Hornung (1986).

Figure 2: Mach reflection in steady state flow. Physical (left) and shock polar (right). Adapted

from Hornung (1986).

The numbers 1 to 5 represent flow regions with different flow conditions.

Region 1 has the free stream conditions, i.e. undisturbed flow. Region 2 has

the flow that has been deflected by the incident shock wave I, by an amount

equivalent to the angle of the wedge. Region 3 has the flow that has been

deflected by the reflected shock wave R. Region 4 has the flow that has

been slowed down to subsonic speed by the Mach stem S. The number 5


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simply represents the point where the Mach stem touches the plane of

symmetry of the reflecting surface. P is the triple point, V is the shear layer

(or the vortex sheet), q1 is the streamline and is the incident shock waves

angle with respect to the plane of symmetry. M=1 is the Mach number

equaling the local speed of sound when the flow in region 4 is accelerated.

The flow acceleration occurs because as the shear layer curves towards and

then away from the plane of symmetry, a converging-diverging nozzle is

formed.

The shock polars in figures 1 and 2 represent what happens physically by

relating the pressures in the different regions to the deflection angles the flows

in each of the regions go through. P represents a regions pressure and P1

represents the pressure in region 1. Therefore the vertical axis is the natural

log of the ratio of the local region to that of region 1. is the deflection angle

of the flow in a region. The deflection angle of region 1 is zero because the

flow in this region is at free stream conditions. The deflection angles of the

other regions are determined relative to that of region 1. Note that the

deflection angles and pressures of regions 3 and 4 is the same. This so

because both pressure and deflection angle do not change across the shear

layer.

Flow speed behind the reflected shock wave in a regular reflection pattern is

still supersonic, whereas the flow field behind the reflected shock wave in a

Mach reflection pattern is also supersonic, but the flow field behind the Mach

stem is subsonic.

Depending on the reflection pattern that results from the shock waves

encounter with the reflecting surface, the flow speed behind the reflection

configuration could be supersonic or a mixture of supersonic and subsonic,

with a shear layer separating the two.

Since the reflection pattern can be affected by the flow speed, orientation of

the reflecting surface, the angle of the incident or incoming shock wave, then


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if any of those parameters were to change after the reflection pattern has

been established, the reflection pattern could change from say regular

reflection to Mach reflection or vice versa. The nature and the conditions

under which this change or transition in reflection pattern is of importance in

better predicting the flow speed expected when there are shock wave

reflections in the flow field.

When the shock wave happens to be a strong shock wave and with the

combination of the above mentioned parameters, one either gets a shock

wave angle that is less than the von Neumann angle, greater than the

detachment angle or falls in a region between the two angles. The von

Neumann and detachment angles are represented in figures 3 and 4

respectively.

Figure 3: A shock polar diagram showing the condition at which the shock wave angle is the vonNeumann angle.


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Figure 4: A shock polar diagram showing the condition at which the shock wave angle is the

detachment angle (solid line) or the sonic angle (dashed line).

Ben-dor et al. (2001) describe the von Neumann (also known as the

mechanical equilibrium) angle as occurring when the reflected shock polar

intersects the vertical axis at exactly the normal shock point of the incident

shock polar. The von Neumann angle marks the shock wave angle abovewhich if one were to keep the Mach number constant whilst increasing the

shock wave angle, one moves to a case where both regular and Mach

reflections are possible from a case where only regular reflection is possible.

Ben-dor et al. (2001) also describe the detachment angle as occurring when

the reflected shock polar is tangent to the vertical axis. The detachment angle

marks the shock wave angle below which if one were to keep the Mach

number constant whilst decreasing the shock wave angle, one moves to acase where both regular and Mach reflections are possible from a case where

only Mach reflection is possible. It can be seen from figure 4, that the

detachment and sonic angles are very close to each other. So for all practical

purposes the detachment and sonic angles are treated as one.

The above description is best illustrated in figure 5.

When the shock wave angle is less than the von Neumann angle, one gets

regular reflection of the shock wave, reflecting off the surface, and if the shock


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wave is at greater than the detachment angle one gets Mach reflection of the

shock wave. If the shock wave falls in the region between the two angles, one

either gets regular reflection or Mach reflection. The region between the two

angles is called the dual solution domain. The above description is illustrated

in figure 5 where a shock wave angle is plotted against Mach number.

What figure 5 indicates is that when you start with regular reflection and then

keep the Mach number constant while increasing the angle that the flow must

turn, one will move into the dual solution domain, but the reflection pattern will

remain regular.

For the case where one started with Mach reflection but kept the Mach

number constant and reduced the angle that the flow must turn, one would

move into the dual solution domain yet remaining with the Mach reflection

pattern.

What one can also deduce from the figure 5 is if one were to keep constant

the angle that the flow must turn through, but increase the free-stream Mach

number, which decreases the shock wave angle, one would move into the

dual solution domain whether one started with regular or Mach reflection, but

would remain with the reflection pattern that one started with.

Unfortunately this is only true if one assumes that the flow field is two-

dimensional, whereas in reality the flow field is generally three-dimensional.

The two-dimensional flow field to which figure 5 is applicable can be

approximated.

For three-dimensional flow fields one looks at the span of the body

encountered by the flow, the span being in the transverse direction to the

stream-wise direction. If the span of the body is small as compared to the

length of the body in the stream-wise direction, three-dimensional effects

creep in and the predictions of figure 5 may not necessarily be true.

But as mentioned above, two-dimensional flow fields can be approximated.

This can be achieved by having a span that is much bigger than the length,

where the flow in the middle of the body is essentially two-dimensional with


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three-dimensional effects not having any effect. But three-dimensional effects

will still affect the flow near the edge of the bodys span.

Figure 5: The von Neumann and detachment criteria as functions of Mach number for a specific-

heat ratio of 1.4, where d is the detachment criterion angle, N is the von Neumann criterionangle and is the incident shock wave angle.


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Background and literature review

Henderson & Lozzi (1975) found that the detachment criterion for transitionwas wrong for every flow that they investigated in detail, and these flows

include steady, pseudo-steady and unsteady cases. Skews (2000) also

support this conclusion, but went a bit further by investigating the flow from a

three-dimensional point of view. Skews (2000) took into consideration the

effects of the finite wedge edges. The experimentation conducted by Skews

(2000) also supported the dual solution phenomenon. Most researchers in

their results would observe the dual solution, where they concluded that eitherRR or MR could occur. But Skews (2000) showed that both types of

reflections, i.e. the RR and MR do occur at the same time in the same flow.

One of the aims of the present study is based on the work done by Skews

(2000).

Figure 6 shows the reflection transition phenomena observed by Skews

(2000).

Figure 6: An oblique shadowgraph picture showing transition from regular reflection to Mach

reflection, taken by Skews (2000).


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Irving-Brown and Skews (2004) also show experimental results of the same

wedge arrangement as that of figure 6, and these results are shown in figures

7 and 8.

Figure 7: Conventional shadowgraph showing regular reflection. Adapted from Irving-Brown

and Skews (2004).

Figure 8: Orthogonal shadowgraph corresponding to figure 7. Adapted from Irving-Brown and

Skews (2004).

Ivanov et al. (1999) used a laser sheet vapour screen technique to visualise

the 3D structure of a shock wave reflection in wind tunnel experiments with

symmetrical wedges. A new shock reflection configuration was observed. For

this configuration when moving along the span-wise direction, the Mach

reflection existing in the central plane is changed into regular reflection and

later again the peripheral Mach reflection appears. Figure 9 shows the

reflection transition phenomena observed by Ivanov et al. (1999).


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Figure 9: Laser sheet images at different span-wise positions ( z) taken by Ivanov et al. (1999).

Sudani et al. (2002) conducted experiments for an asymmetric arrangement

and also managed to observe the same phenomena observed by Ivanov

(1999), and these observations are shown in figure 10.

Figure 10: Schlieren and vapour-screen pictures in the asymmetric arrangement. M=3.0.

Adapted from Sudani et al. (2002).

Sudani et al. (1999) conducted experimental studies of shock wave reflections

in steady flows at Mach numbers of 3 to 4 in a blow-down supersonic wind

tunnel. In a symmetric arrangement where the upper wedge is vertically

moved with its deflection angle fixed, the transition to Mach reflection occurred

at a certain location when the inlet aspect ratio was increased, hence no

significant effect of inlet aspect ratio on the transition location could be

observed. Vapour screen visualisation technique was used for the

experimental studies and it was found that the Mach stem has its maximum

height at the span-wise centre and that three-dimensional effects promote

regular reflection. Their experimental data led to the hypothesis that wind

tunnel disturbances cause the transition to Mach reflection in the dual-solution

domain.


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However Kudryavtsev et al. (2002) have concluded, through the use of

numerical simulations, in the dual solution domain Mach reflection was more

stable than regular reflection. They established this through the use of what

they call free-stream disturbances.

As for the effects coming from downstream flow conditions on the Mach stem

height, Chpoun and Leclerc (1999) show that there is none. This is illustrated

in figure 11 where experiments were conducted at hypersonic conditions on

wedges with various trailing edge corner angles. The figure shows wedges

with the same wedge angle, but the trailing edge corner angle varies from 45

up to 145. As is pointed out in Chpoun and Leclerc (1999) this observed

phenomenon contradicts some analytical findings.

Hornung (1986) concludes that the expansion wave coming from the trailing

edge of the wedge causes the slip stream to curve away from the horizontal

symmetry plane, thereby forming the diverging-converging nozzle. Because

up until the throat of the nozzle the flow is subsonic, Hornung (1986)

concludes that an information pathway is created back to the triple point.

Henderson et al. (2001) make the point that by definition regular reflection has

no boundary layer interaction, therefore a shock wave reflection off a plane of

symmetry will always give either regular reflection or Mach reflection. On the

contrary, precursor regular reflection always appears with boundary layer

interaction, as on a ramp surface.


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Figure 11: shadowgraph pictures showing the absence of influence on the Mach stem height by

downstream flow conditions. Adapted from Chpoun and Leclerc (1999).


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Objectives

The first aim of this project is to carry out simulations of experiments

conducted independently by Ivanov et al. (1999) and Sudani et al. (1999) on

shock wave interaction of two incident shock waves generated by two

symmetrically arranged wedges. The wedges have a high aspect ratio, i.e. the

ratio of the width to the length, which eliminates three-dimensional effects at

the model centre line. For this setup, the shape of the reflected shock waves

at the model centre is not curved, but flat.

The reflection pattern observed by both teams is a Mach reflection at the

model centre line and regular reflection as one moves towards the edge of the

model. However, further out the reflection pattern transitions to Mach

reflection and remains so, as expected, with a peripheral Mach reflection

pattern.

The second aim is to conduct simulations at lower Mach numbers with the

intent to reproduce the reflection patterns observed by the two above-

mentioned teams. Thereafter experiments will be conducted at Mach 3.3 inthe supersonic wind tunnel at University of the Witwatersrand.

The third aim is to analyse the complete flow structure from both the

experimental and computational simulation point of view, to determine the

effect of changing the geometrical set-up on the reflection pattern.

The fourth aim is numerically resolve the sudden change in height of the

Mach stem at the transition point from regular reflection to Mach reflection

observed by Skews (2000).


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Method and procedure

The research project follows work done by the candidate on a fourth year

research project in which the fourth aim of this proposal was the main aim.

The title of the fourth year research project was The Numerical investigation

of the instability of the mechanical equilibrium point. Although retrospectively

this title is incorrect since the talk of a mechanical equilibrium point only

applies when one assumes a two-dimensional flow field. The method used for

the investigation for this fourth year project was a commercial Computational

Fluid Dynamics (CFD) software package named STAR-CD.

Subsequent to the fourth year project, preliminary work had been done in

investigating some of the objectives of the current project. In this preliminary

work Fluent, another commercial CFD software package was used.

These two works were used as the basis from which this current project would

be carried out.

Case descriptions

The above stated objectives describe different shock wave reflection

phenomena occurring under different free stream Mach numbers and

geometric configurations. The first and second objectives were investigated

using a case with a geometric configuration having big transverse dimensions

relative to those in the streamwise direction. This case meant for the

investigation of the first and the second objectives will be called Double

transition Case.

The fourth objective was investigated using a case with a geometric

configuration having small transverse dimensions relative to those in the

stream wise direction. This case meant for the investigation of the fourth

objective will be called Single transition Case.

Each of these cases will be described separately in the following sections.


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Single transition Case

The geometric setup in the investigation of transition from regular reflection to

Mach reflection consists of two symmetrically arranged wedges with finite

spans with one placed above the other. This arrangement results in the same

reflection outcome as would have been achieved with using a wedge and a

flat surface parallel to the horizontal. But the advantage of using this outlined

setup is that one completely removes the affect of a boundary layer on the

reflection pattern. Hence one has a virtually perfect adiabatic surface.

Because the wedges are symmetrically arranged about the perfect reflection

surface, one can assume symmetry and use only one wedge. Symmetry also

exists about the centre plane of the wedge span; hence half a wedge is usedin the CFD simulations.

For this case where the span of the wedge is small in comparison to the

length of the wedge, in this case span to length ratio, called the aspect ratio,

of 0.5 ( a span of 20 mm and a length of 40 mm), the trailing edge gap is 9

mm with a wedge angle of 25. This case is a numeri cal replication of work

carried out by Skews (2000) and Skews et al. (2004), where a blow-down

wind tunnel with a cross section of 100 mm x 100 mm was used at M = 3.1,

hence the tunnel wall is modelled to be 50 mm away from the centre of the

wedge.

Pictures showing the geometric setup of the wedges are shown in Figure 12.


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Figure 12: Pictures showing arrangements of the two wedges used in the Single transition case.

Double transition case

The geometric configuration for this case is the same as that for the single

transition case.

For a case where the span of the wedge is large in comparison to the length

of the wedge, the aspect ratio is 3.75 with a span of 300 mm and a length of

80 mm, the trailing edge gap of 24 mm and a wedge angle of 21.4. This case

is a numerical simulation of the work carried out by Ivanov et al. (1999) using

a wind tunnel with a cross section of 600 mm x 600 mm at free stream Mach

number of 4. The tunnel wall was modelled to be 300 mm away from the

centre of the wedge.

Pictures showing the geometric setup of the wedges for this case are shown

in Figure 13.

Isometric view Side view

Front view


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Figure 13: Pictures showing arrangements of the two wedges used in the Double transition case.

Detailed drawings showing dimensions of the wedge setups for both cases

are in Appendix C.

Fluent and Starccm+, two commercial CFD codes, were used for the

simulations. In all the simulations inviscid flow was assumed, except in one

instance for the single transition case where a laminar flow was assumed in

order to evaluate whether there are any viscous effects. The mesh generated

in Gambit was imported into both Fluent and Starccm+, but refinement was

done using Fluent. This refinement was done based on density gradients, and

it was done until refinement could be done any longer. The refined mesh was

then also used in Starccm+ although without the benefit of an initial solution to

start from. For both Fluent and Starccm+ a coupled (density based) solver

was selected. This is because coupled solvers are very good at resolving

Isometric view Side view

Front view


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discontinuities like shock waves in flows. For Fluent, a second-order upwind

spatial discretisation scheme with a Roe-Flux Difference splitting (FDS) limiter

was used because it works well in flows with discontinuities. Algebraic Miltigrid

method with V-cycle was used to speed up the simulation while not affecting

accuracy of the results. Adaptation was done resulting in over 1.84 million

cells for the Single transition Case and over 1.7 million cells for the Double

transition Case.

All the codes were ran on 64-bit Linux servers, with any server having up to 8

GB of RAM, and the option of parallelising the simulations on up to 24

computers.


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Discussion of results

Double transition Case

For the case with a wedge aspect ratio of 3.75, the results presented are fromboth Fluent and Starccm+ although the former codes results are for a Mach

number of 4.0 and the latter codes results are for Mach numbers of 3.1 and

2.9.

As stated in the first objective in the objectives section, Ivanov et al. (1999)

experimentally observed a very interesting reflection phenomenon, wherethere is Mach reflection observed at the centre of the wedge which transitions

to regular reflection as one moves towards the tunnel wall. But as expected

the regular reflection transitions to a peripheral Mach reflection as one move

further towards the tunnel wall. Ivanov et al. (1999) ran their experiments at

Mach number 4.0.

The shock wave reflection transition phenomenon is very interesting because

in the literature the focus is on establishing the transition criterion from regular

reflection to Mach reflection and vice versa. And this is done from the

perspective of the flow being view or assumed to be two dimensional or

pseudo-two dimensional. From this observed shock wave reflection transition

phenomenon (i.e. observed by Ivanov et al. (1999)) it can be seen that

although the flow is essentially two dimensional at the centre of the geometric

setup, the reflection pattern changes from Mach reflection at the centre as

one moves outwards from the centre. The Mach surface (Mach stem in two

dimensions) narrows in height up to the point where there is transition from

Mach reflection to regular reflection. At this point at which this transition from

Mach reflection to regular reflection occurs, the incident shock wave is still flat

and not curved. As one moves further away from the centre the regular

reflection persists for a short distance and then there is the expected

transition to Mach reflection. Both of these transitions are smooth, which might

suggest the stability of the reflection patterns.


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It should be mentioned that the Mach reflection at the centre of the geometric

setup is as predicted by two-dimensional theory. Then a question naturally

arises of why is there transition from Mach reflection to regular reflection if the

Mach reflection is as predicted by two dimensional theory, meaning that if all

conditions are kept constant then the reflection pattern should be stable and

not change. This question could be answered by viewing this as being further

evidence that three dimensional effects promote regular reflection over Mach

reflection as concluded by Sudani and Hornung (1998).

It is worth noting that in the literature there is no transition criterion for a

three-dimensional transition.

As stated above the experiment conducted by Ivanov et al. (1999) was

modelled using two commercially available CFD codes. The modelling was

conducted at the Mach numbers 4 (the Mach number for the experiment), 3.1

and 2.9. The aim for running simulations at Mach numbers 3.1 and 2.9 was to

achieve what is stated in the second objective. The results for the simulation

at Mach number 4 are presented in figure 14 below. They clearly agree with

the experimental observation made by Ivanov et al. (1999) at those marked

stations, presented in figure 15. But the only difference between the two set of

results is that the simulation was run as an unsteady one, where the

presented results with a double transition fade away and one gets a case with

regular reflection at the centre that transitions to the expected peripheral

Mach reflection.

This brings one back to the conclusion made by Sudani and Hornung (1998)

that three dimensional effects do promote regular reflection over Mach

reflection.

But then when one runs the simulations at Mach numbers 3.1 and 2.9 one

observes the double transition phenomenon with the simulations being run in

a steady state. The results at Mach numbers 3.1 and 2.9 are presented in

figures 16 and 17 respectively at the same stations as those of the experiment

conducted by Ivanov et al. (1999). The results from the simulations at these

two Mach numbers satisfy the second objective of observing the double

transition at Mach numbers attainable at the Wits Flow Research Unit facility.


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Figure 14: CFD pictures at span-wise positions (z) of, from left to right 0 mm, 158 mm and 198

mm, obtained using Fluent for M

= 4.0.

Figure 14: Laser sheet Images at different span-wise positions ( z) taken by Ivanov et al. (1999)for M

= 4.0.


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Figure 16: CFD pictures at span-wise positions (z) of, from top to bottom 0 mm, 158 mm and 198


= 3.1.


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Figure 17: CFD pictures at span-wise positions (z) of, from top to bottom 0 mm, 158 mm and 198


= 2.9.

In all the simulations inviscid flow was assumed, except in one instance for

the single transition case where a laminar flow was assumed in order to

evaluate whether there are any viscous effects. The mesh generated in

Gambit was imported into both Fluent and Starccm+, but refinement was done

using Fluent. The refined mesh was then also used in Starccm+ although

without the benefit of an initial solution to start from as in Fluent.


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Figure 18: A flooded plan view of Mach number plotted on the horizontal symmetry plane. Thisis for M

= 3.1.

Figure 18 shows a plan view plot of Mach number from which one can see the

subsonic region behind the flat portion of the Mach surface. As stated earlier,

for this flat portion two-dimensional theory can predict the reflection pattern.

But one can also see the transition from Mach reflection to regular reflection

within a small part of the flat portion. This transition can not be predicted by

two-dimensional theory. Because the flow coming at the incident shock wave

is perpendicular (in a stream wise sense) to the shock wave, and the flow

behind the shock wave is supersonic, it can be seen that the reflection pattern

is indeed regular.

From Figure 18 one can also see the extent of the subsonic region behind the

Mach surface. This subsonic region has the converging-diverging nozzle

effect in both the stream wise and lateral direction. The shape of the slip


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stream bounding the subsonic region can also be observed from figure 18.

There is hardly any change in speed behind the subsonic region.

Figure 19: A flooded plan view of pressure plotted on the horizontal symmetry plane. This is for

M

= 3.1.

From figure 19 one can see that there is little difference in pressure as one

move across the peripheral Mach surfaces. Another observation is that

because there is no change in pressure across the slip stream (in three-

dimension it becomes a slip surface, refer to figures 20 and 24 for the shape),

the shape of the subsonic region as seen in figure 18 can not be observed in

figure 19. Instead a strange pressure distribution is observed that can not be

correlated with the observed Mach number distribution plotted in figure 18.


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Figure 20: A plan view plot of a slip surface (shear layer) with streamlines moving from right to

left plotted on the horizontal symmetry plane, with the wedge in the background. This is for M= 3.1.

Figure 20 shows how the flow as represented by the streamlines gets affected

by the shock system. Figure 20 also shows the plan view of the slip surface

(shear layer).


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Figure 21: A plan view plot of entropy on the horizontal symmetry plane. This is for M

= 3.1.

Figure 21 shows the increase or production in entropy in the flow behind the

incident shock wave and the Mach surface. From figure 21 one can see that

the highest production of entropy occurs in the subsonic region. Since the

subsonic region is behind the Mach surface, this means there is more energyloss in the flow going through the Mach surface than through two oblique

shock waves. This is the only explanation since another way in which entropy

could be increased or created is by turbulence, but the flow is modelled as

being inviscid. The small region where there is regular reflection the flow goes

through the incident and reflected shock waves, but the flow behind them has

less turbulence than the subsonic region. Explained another way, if one were

to look at the normal components of the incident and reflected shock waves

their combined Mach number drop across them is smaller than that across a

Mach stem, with the Mach stem being a normal shock wave.

The peripheral Mach surface produces the least amount of entropy because

the flow comes at it at an angle. The four small red regions of high entropy are

due to part of the Mach surface in front of them being stronger than the rest of

the Mach surface and have nothing to do with the transition points. This is

easily verifiable when comparing figures 20 and 21 by observing where the

subsonic region ends as described by the slip surface.


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Figure 22: A front view of Mach=2.8 isosurface showing the double transition phenomenon. This

is for M

= 2.9.

From figure 22 one can observe the shape of the double transition

phenomenon in three dimensions. At the centre of the isosurface is the Mach

surface, though small in size. One can easily see the tapering off of the

central Mach surface as one moves outwards until there is transition to regular

reflection. Then one can definitely see the rapid expansion of the peripheral

Mach surface at the edges.


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Figure 23: A side view of Mach number contours on the vertical symmetry plane for M

= 3.1.

Prediction by two-dimensional theory applies at the centre of the wedge

arrangement. In this case this is illustrated in figure 23 where the Mach

reflection observed in the figure is expected, because for this wedge angle

and Mach number the reflection pattern falls within the dual solution domain.


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View from the wedge setup inlet

Isometric view of the shockwave reflection

Figure 24: Starccm+ images, for the halves of the two wedges, of an isosurface of Mach 2.9, for

M

= 3.1.


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Plan view of the slip surface

Isometric view of the slip surface

Figure 25: Starccm+ images, for half a wedge, of an isosurface of Mach 1, showing the shape of

the slip surface for M

= 3.1.

Figure 24 shows the shape of the double transition phenomenon, but in this

instance for M = 3.1. The associated slip stream surface is shown in figure

25.


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Single transition case

As in the double transition case the single transition case simulations were

run using both Fluent and Starccm+, but results for a particular flow feature

will be presented using pictures from only one of the two packages. Additional

pictures of the results will be presented in Appendix B.

Just as it was discussed in the experimental and computational setup sections

above, the single transition case objective was examined using a wedge with

an aspect ratio of 0.5 and the simplifying assumption made for the double

transition case were applied in this case too.

In as far as answering the fourth aim put forward in the objective section, the

sudden change in height of the Mach stem at the transition from regular

reflection to Mach reflection can not be observed from the simulations of

either of the two simulations packages. The reason for this could be that since

the wind tunnel used by Skews (2000) was noisy, vibrations in the flow could

have triggered the sudden transition from regular reflection to Mach reflection.

This observation is presented in figures 26 and 27 showing the experimental

observation made by Skews (2000) and the simulation results obtained by this

author using Starccm+, respectively.


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Figure 26: An oblique shadowgraph picture showing transition from regular reflection to Mach

reflection, taken by Skews (2000).

Figure 27: A 3-D Starccm+ projected view of a Mach number isosurface at Mach = 2.9, showing

transition from regular reflection to Mach reflection.


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Figure 28:A 3-D Starccm+ projected view of a Mach number isosurface at Mach = 2.9, showingtransition from regular reflection to Mach reflection. Simulation ran with a laminar flow.

Figure 29: A close up of the transition from regular reflection to Mach reflection for figure 26.


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Figure 30: A close up of the transition from regular reflection to Mach reflection for figure 27.

Figure 28 shows the same view as figure 27, but for a laminar flow. The aim of

running a laminar simulation was to see if the sudden jump in the transition

from regular reflection to Mach reflection observed by Skews (2000) might be

due to viscous effects. As can be from both figures 27 and 28, there is no

difference in the flow patterns. Close ups of both figures are presented in

figures 29 and 30 respectively.


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Figure 31: A flooded plan view of Mach number plotted on the horizontal symmetry plane. Thisis for M

= 3.1.

Figure 31 shows a flooded plan plot of Mach number on the horizontal

symmetry plane for the single transition case. As one can observe the incident

shock wave is curved with supersonic flow behind the oblique shock wave.

All this is well illustrated in figure 32 below where a flooded plot of Mach

number is plotted on the vertical symmetry plane.

As mentioned earlier figure 31 is a flooded plot of Mach number plotted on the

vertical symmetry plane, whereby Starccm+ is used as the CFD solver.


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Figure 32: A flooded plan view of Mach number plotted on the vertical symmetry plane. This is

for M = 3.1.


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Figure 33: A flooded plan view of density plotted on the horizontal symmetry plane. This is forM

= 3.1.

Figure 34: Orthogonal shadowgraph showing the single transition phenomenon. This picture is

the same figure 8, but rotated by 180 for easy comparison with figure 29. Adapted from Brown

and Skews (2004).


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Figure 35: A flooded plan view of density plotted on the vertical symmetry plane. This is for M

= 3.1.


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Figure 36: A flooded plan view of pressure plotted on the vertical symmetry plane. For M

= 3.1.

Figure 37: A front view of Mach=2.9 isosurface showing the right half of the single transitionphenomenon with the vertical plane of symmetry to the left. This is for M

= 3.1.


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Figure 38: A close-up of Figure 37.

The shape of the transition is shown in figures 37 and 38. From both figures it

can be seen that the transition is smooth and not sudden contrary to the

experimental observation by Skews (2000).


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Figure 39: A front view of Mach=2.9 isosurface showing the lower half of the single transition

phenomenon below the horizontal plane of symmetry. This is for M

= 3.1.


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Figure 40: A plan view showing streamlines paths being affected by going through shock waves.

For M = 3.1.

The effect that the shock waves have on the flow is easily illustrated with the

use of streamlines as in figure 40. From the figure it is seen that with flow

coming from the right, the flow is generally deflected towards the tunnel wall

(being the top of the figure.) This effect is well pronounced in figure 41

showing how the flow moves towards the tunnel walls and away from the

wedge surface.


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Figure 41: A plan view showing streamlines paths being affected by going through shock waves,

for both sides of the wedge. For M = 3.1.


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Figure 42: A side view showing streamlines paths being affected by going through shock waves.

For M

= 3.1. streamlines released just below the horizontal plane of symmetry.

From figure 42 one can see the effect that the incident shock wave has on the

flow by observing that the streamlines are deflected upwards towards the

horizontal symmetry plane. This is as expected as predicted by two-

dimensional theory as to the trajectory of streamlines going through the shock

wave. This effect is also seen in figure 43. Because figure 42 is viewed from

the vertical symmetry plane, what is observed in the figure is the effect due to

the regular reflection pattern. But the effect due to both the regular reflection

and Mach reflection is observed in figure 43. The already observed upward

deflection of the flow by the regular reflection pattern is to the right of the

figure, whereas the observed zero change in height of the streamlines to the

left of the figure is due to the Mach reflection pattern. Again this is as

expected as predicted by two-dimensional theory. One can conclude that two-

dimensional theory could be used in three-dimensional flow fields at different

stations transverse to the stream wise direction, and the theory would be validfor that particular station.


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Figure 43: A front view showing streamlines paths being affected by going through shock waves.

For M

= 3.1.


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Figure 44: An isometric view showing streamlines paths being affected by going through shockwaves. For M

= 3.1.


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Conclusions

The sudden transition from regular reflection to Mach reflection observed

experimentally by Skews (2000) could not be observed numerically whenusing two commercial CFD codes, a possible reason for this being the noisy

wind tunnel used by Skews (2000). The reflection phenomenon observed

experimentally by Ivanov et al. (1999) was replicated numerically using the

two above mentioned commercial CFD codes for the same Mach number of 4

and a lower Mach numbers of 3.1 and 2.9, although for Mach 4 the simulation

was unsteady.


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References

1. L. F. Henderson, A. Lozzi (1975), Experiments on transition of Mach

reflection, J. Fluid Mech., Vol. 68, pp. 139-155.

2. M. S. Ivanov, G. P. Klemenkov, A. N. Kudryavstev, S. B. Nikiforov, A. A.

Pavlov, A. M. Kharitonov, V. M. Fomin (1999), Wind tunnel experiments on

shock wave reflection transition and hysteresis, 22nd International

symposium on Shock waves, Imperial College, London, UK, paper 1121.

3. B. W. Skews, J. A. Mohan, N. Menon (2004) Unexpected wave patterns in

a non-circular supersonic duct inlet. 4th South African Conference on

Computational and Applied Mechanics, SACAM 2004, Mist hills,Johannesburg, South Africa.

4. B. W. Skews (2000), Three-dimensional effects in wind tunnel studies of

shock wave reflection, J. Fluid Mech., Vol. 407, pp. 85-104.

5. N. Sudani, M. Sato, T. Karasawa, A. Tate, J. Noda, M. Watanabe, Y.

Mizobuchi, S. Hamamoto, Effects of three-dimensionality and asymmetry on

transition to Mach reflection, 22nd International symposium on Shock waves,

Imperial College, London, UK, paper 1999.

6. A. Chpoun, E. Leclerc (1999), Experimental investigation of the influence

of downstream flow conditions on Mach stem height, Shock Waves 9, pp.

269-271.

7. A.N. Kudryavtsev, D.V. Khotyanovsky, M.S. Ivanov, A. Hadjadj, D.

Vandromme (2002), Numerical investigations of transition between regular

and Mach reflections caused by free-stream disturbances, Shock Waves 12,pp. 157-165.

8. H. Horning (1986), Regular and Mach reflection of shock waves, Ann.

Rev. Fluid Mech. 18, pp. 33-58.

9. L. F. Henderson, K. Takayama , W. Y. Crutchfield & S. Itabashi (2001),

The persistence of regular reflection during strong shock diffraction over rigid

ramps, J. Fluid Mech., vol. 431, pp. 273-296.

10. G. Ben-Dor, O. Igra, T. Elperin (2001), Handbook of Shock Waves, vol. 2.


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11. Y.A. Irving-Brown, B.W. Skews (2004), Three-dimensional effects on

regular reflection in steady supersonic flows, Shock Waves 13, pp. 339349.


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Appendix A: Additional pictures for the double

transition case.

Figure 44: A flooded plan view of Mach number plotted on the horizontal symmetry plane. This

is for M

= 2.9

Figure 45: A flooded plan view of Pressure plotted on the horizontal symmetry plane. This is for

M

= 2.9


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Figure 46: A contoured side view of Pressure plotted on the vertical symmetry plane

superimposed with the slip surface, showing the shape of the slip surface viewed from the side.

This is for M

= 3.1


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Appendix B: Additional pictures for the singletransition case.

Figure 47: A flooded plan view of Pressure plotted on the horizontal symmetry plane.


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Figure 48: A side view showing streamlines (in colour) paths being affected by going through

shock waves. For M

= 3.1. Streamlines released just below the horizontal plane of symmetry.

Figure 49: A front view showing streamlines paths (in colour) being affected by going through

shock waves. For M

= 3.1.


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Appendix C: Detailed dimensions of the wedge setup

for each test case.


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dimensions are in millimeters

Sketches with dimensions fordouble transition case

482

1.40

C

2 31 4

B

A

D

E

F

WEIGHT:

A4

SHEET 1 OF 1SCALE:1:4

DWG NO.

TITLE:

REVISIONDO NOT SCALE DRAWING

MATERIAL:

DATESIGNATURENAME

DEBUR AND

BREAK SHARP

EDGES

FINISH:UNLESS OTHERWISE SPECIFIED:

DIMENSIONS ARE IN MILLIMETERS

SURFACE FINISH:

TOLERANCES:

LINEAR:

ANGULAR:

Q.A

MFG

APPV'D

CHK'D

DRAWN

300

225


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Dimensions are in millimeters

4

0

60

C

2 31 4

B

A

D

E

F

WEIGHT:

A4

SHEET 1 OF 1SCALE:1:1

DWG NO.

TITLE:

REVISIONDO NOT SCALE DRAWING

MATERIAL:

DATESIGNATURENAME

DEBUR AND

BREAK SHARP

EDGES

FINISH:UNLESS OTHERWISE SPECIFIED:

DIMENSIONS ARE IN MILLIMETERS

SURFACE FINISH:

TOLERANCES:

LINEAR:

ANGULAR:

Q.A

MFG

APPV'D

CHK'D

DRAWNSketches and dimensions for theSingle transition case

18

25


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MSc_Eng_-Steady Three-dimensional Shock Wave Reflection Transition Phenomena

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